Recent diffusion models provide a promising zero-shot solution to noisy linear inverse problems without retraining for specific inverse problems. In this paper, we reveal that recent methods can be uniformly interpreted as employing a Gaussian approximation with hand-crafted isotropic covariance for the intractable denoising posterior to approximate the conditional posterior mean. Inspired by this finding, we propose to improve recent methods by using more principled covariance determined by maximum likelihood estimation. To achieve posterior covariance optimization without retraining, we provide general plug-and-play solutions based on two approaches specifically designed for leveraging pre-trained models with and without reverse covariance. We further propose a scalable method for learning posterior covariance prediction based on representation with orthonormal basis. Experimental results demonstrate that the proposed methods significantly enhance reconstruction performance without requiring hyperparameter tuning.

翻译：近期扩散模型为含噪线性逆问题提供了一种无需针对特定逆问题重新训练的有前景的零样本解决方案。本文指出，现有方法可被统一解释为：对难以处理的去噪后验分布采用手工设计的各向同性协方差高斯近似，以逼近条件后验均值。受此发现启发，我们提出通过采用基于最大似然估计确定、更具理论依据的协方差矩阵来改进现有方法。为实现无需重新训练的后验协方差优化，我们基于两种专门设计的方案提供了通用即插即用解决方案，分别适用于利用具备与不具备反向协方差信息的预训练模型。我们进一步提出一种基于正交基表示的可扩展方法，用于学习后验协方差预测。实验结果表明，所提方法在无需超参数调优的情况下，显著提升了重建性能。